The following data is a sample of daily maximum
temperatures in New York in March (from 2006-2008).
a. Calculate the mean (1 decimal place) and
standard deviation (2 decimal places) of this
data. You may use technology to answer this
question. Only the final answer is required. [3]
b. Determine a reasonable interval size and number
of intervals. Produce a properly labeled
histogram for the grouped data using technology.
Paste this graph into your solutions. [4]
c. In March, the temperatures in Seattle are
normally distributed with a mean daily
maximum temperature of 4.5ºC and a standard
deviation of 6.25ºC. What percent of days
would you predict would be between 0ºC and
10ºC? This question must be answered algebraically
I have already done part a) mean = 3.61 ; standard deviation 5.9 and part b) graph.
But I am unsure how to answer part (c)
c. 0 to 10 is within +- one standard deviation. Doesn't that cover 68 percent of the days?
yes I got that through looking at the graph but I need to show it algebraically ....so I think we have to look at the Z values
z = (x - mean)/sd
that's what I am trying to figure out
To answer part (c), you need to use the concept of standard scores (also known as z-scores) in order to convert the given temperatures into their corresponding z-scores. Then, you can use the standard normal distribution table or a statistical calculator to find the cumulative probability between the specified temperatures.
The formula to calculate the z-score is:
z = (x - μ) / σ
Where:
- z is the z-score
- x is the value you want to convert to a z-score
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
Let's calculate the z-scores for the temperatures 0ºC and 10ºC in Seattle:
For 0ºC:
z1 = (0 - 4.5) / 6.25
For 10ºC:
z2 = (10 - 4.5) / 6.25
Now, we can use a standard normal distribution table or a statistical calculator to find the cumulative probability associated with these z-scores. The cumulative probability represents the area under the curve between the specified z-scores.
If using a standard normal distribution table, you would look up the z-scores in the table and subtract the cumulative probabilities to find the percentage:
P(0ºC ≤ X ≤ 10ºC) = P(z1 ≤ Z ≤ z2) = P(Z ≤ z2) - P(Z ≤ z1)
For example, let's say you find that P(Z ≤ z1) is 0.30 and P(Z ≤ z2) is 0.70. Then, you would calculate:
P(0ºC ≤ X ≤ 10ºC) = 0.70 - 0.30 = 0.40
Therefore, you would predict that approximately 40% of days in March would have temperatures between 0ºC and 10ºC in Seattle.
Remember to substitute the actual z-scores that you calculate into the formula to get the final result.